Just starting out-question

I've been recently practicing on an old electric bass. I feel most comfortable with my right hand on the neck, and left had plucking. I want to get more serious and I'm looking to purchase a fender jazz bass, early 70s...But, there are so few "left handed basses" do you suggest to get a right handed bass, and then switch the strings? Is that possible? And does it make it an issue when it comes to reading music, if I played a bass with my right hand on the neck?